Long-time asymptotics for fully nonlinear homogeneous parabolic equations

We study the long-time asymptotics of solutions of the uniformly parabolic equation $$ u_t + F(D^2u) = 0 \quad{\rm in}\, {\mathbb{R}^{n}}\times \mathbb{R}_{+},$$ for a positively homogeneous operator F, subject to the initial condition u(x, 0) =  g(x), under the assumption that g does not change sign and possesses sufficient decay at infinity. We prove the existence of a unique positive solution Φ⁺ and negative solution Φ^?, which satisfy the self-similarity relations $$\Phi^\pm (x,t) = \lambda^{\alpha^\pm}\Phi^\pm ( \lambda^{1/2} x,\lambda t ).$$ We prove that the rescaled limit of the solution of the Cauchy problem with nonnegative (nonpositive) initial data converges to ${\Phi^+}$ ( ${\Phi^-}$ ) locally uniformly in ${\mathbb{R}^{n} \times \mathbb{R}_{+}}$ . The anomalous exponents α⁺ and α^? are identified as the principal half-eigenvalues of a certain elliptic operator associated to F in ${\mathbb{R}^{n}}$ .     

Asymptotics of solutions of nonlinear parabolic equations

This paper is concerned with the large time behaviour of solutions to the Cauchy problem of the following nonlinear parabolic equations:

     

Classical solutions of fully nonlinear parabolic equations

In this paper, we study the classical solutions of the fully nonlinear parabolic equation u_t-F(D_x²u)=0,{u_{t}-F(D_{x}^2u)=0,} where the nonlinear operator F is locally C ^1,β almost everywhere with 0 < β < 1. The interior C ^2,α regularity of the classical solutions will be shown without the assumption that F is convex (or concave).     

Singular solutions of some nonlinear parabolic equations

We consider the existence and uniqueness of singular solutions for equations of the formu ₁=div(|Du|^p−2 Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2. Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the existence of very singular solutions, i.e. singular solutions such that ∫_|x|≤r u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result. In the case ϕ(u)=u ^q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal. Dedicated to Professor Shmuel Agmon     

On asymptotics of solutions of parabolic equations with nonlocal high-order terms

In the paper, we study the Cauchy problem for second-order differential-difference parabolic equations containing translation operators acting to the high-order derivatives with respect to spatial variables. We construct the integral representation of the solution and investigate its long-term behavior. We prove theorems on asymptotic closeness of the constructed solution and the Cauchy problem solutions for classical parabolic equations; in particular, conditions of the stabilization of the solution are obtained. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 143–183, 2005.     

Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities

In this paper we discuss continuation properties and asymptotic behavior of -regular solutions to abstract semilinear parabolic problems in case when the nonlinear term satisfies critical growth conditions. A necessary and sufficient condition for global in time existence of -regular solutions is given. We also formulate sufficient conditions to construct a piecewise -regular solutions (continuation beyond maximal time of existence for -regular solutions). Applications to strongly damped wave equations and to higher order semilinear parabolic equations are finally discussed. In particular global solvability and the existence of a global attractor for in is achieved in case when a nonlinear term f satisfies a critical growth condition and a dissipativeness condition. Similar result is obtained for a 2mth order semilinear parabolic initial boundary value problem in a Hilbert space .     

Extinction of solutions for some nonlinear parabolic equations

We are dealing with the first vanishing time for solutions of the Cauchy–Neumann problem for the semilinear parabolic equation ∂ _t u − Δu + a(x)u ^q = 0, where a(x) \geqslant d₀exp( - \fracw( | x | )| x |² ) a(x) \geqslant {d_0}\exp \left( { - \frac{{\omega \left( {\left| x \right|} \right)}}{{{{\left| x \right|}^2}}}} \right) , d ₀ > 0, 1 > q > 0, and ω is a positive continuous radial function. We give a Dini-like condition on the function ω which implies that any solution of the above equation vanishes in finite time. The proof is derived from semi-classical limits of some Schr¨odinger operators.     

On long-time asymptotics of solutions of parabolic equations with increasing leading coefficients

Sharp sufficient conditions on the coefficients of a second-order parabolic equation are examined under which the solution of the corresponding Cauchy problem with a power-law growing initial function stabilizes to zero. An example is presented showing that the found sufficient conditions are sharp. Conditions on the coefficients of a parabolic equation are obtained under which the solution of the Cauchy problem with a bounded initial function stabilizes to zero at a power law rate.     

Long time small solutions to nonlinear parabolic equations

A sharp result on global small solutions to the Cauchy problem $$u_t = \Delta u + f\left( {u,Du,D^2 u,u_t } \right)\left( {t > 0} \right),u\left( 0 \right) = u_0 $$ In Rⁿ is obtained under the the assumption thatf is C^1+r forr>2/n and ‖u ₀‖C²(R ⁿ ) +‖u ₀‖W ₁ ² (R ⁿ ) is small. This implies that the assumption thatf is smooth and ‖u ₀ ‖W ₁ ^k (R ⁿ )+‖u ₀‖W ₂ ^k (R ⁿ ) is small fork large enough, made in earlier work, is unnecessary.     

Stabilization of the solutions of nonlinear stochastic parabolic equations

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 12, pp.1630–1636, December, 1989.     

Local behaviour of solutions to doubly nonlinear parabolic equations

We give a relatively simple and transparent proof for Harnack’s inequality for certain degenerate doubly nonlinear parabolic equations. We consider the case where the Lebesgue measure is replaced with a doubling Borel measure which supports a Poincaré inequality.     

Pseudo-almost periodic viscosity solutions of second-order nonlinear parabolic equations

In this paper we generalize the comparison result of Bostan and Namah (2007) [8] to the second-order parabolic case and prove two properties of pseudo-almost periodic functions; then by using Perron’s method we prove the existence and uniqueness of time pseudo-almost periodic viscosity solutions of second-order parabolic equations under usual hypotheses.